The formula
- Impulse from force and time
J = F_avg·Δt- Impulse-momentum theorem
J = Δp = mv_f - mv_i- Average force during a collision
F_avg = m(v_f - v_i) / Δt
What the symbols mean
| Symbol | Meaning | Unit |
|---|---|---|
J | Impulse delivered to the object | N·s (identical to kg·m/s) |
F_avg | Average force over the contact time | N |
Δt | Duration of the contact | s |
Δp | Change in momentum | kg·m/s |
m | Mass of the object | kg |
v_i | Velocity before the interaction | m/s |
v_f | Velocity after the interaction | m/s |
When it applies
- A short interaction where the force varies through the contact and only the average is within reach.
- The mass stays constant, so the entire momentum change shows up as a change in velocity.
- Impulse is a vector. Choose a positive direction before substituting, and let the signs of v_i and v_f follow it.
- Given a force against time graph, the impulse is the area under the curve.
Worked example
Problem. A 0.145 kg baseball arrives at 40 m/s and leaves the bat at 50 m/s in the opposite direction. The bat is in contact with the ball for 1.2 ms. Find the impulse and the average force.
- Choose a positive direction. Take the direction the ball leaves as positive, so v_i = -40 m/s and v_f = +50 m/s.
- Find the velocity change: v_f - v_i = 50 - (-40) = 90 m/s.
- Impulse: J = mΔv = (0.145 kg)(90 m/s) = 13.05 N·s, or 13 N·s to two significant figures.
- Convert the contact time to seconds, Δt = 1.2 × 10⁻³ s, and divide: F_avg = 13.05 / 0.0012 = 1.1 × 10⁴ N.
Answer. The impulse is 13 N·s and the average force during contact is about 1.1 × 10⁴ N.
Common mistakes
- Subtracting speeds instead of velocities. A ball that reverses direction undergoes a velocity change larger than either speed on its own.
- Leaving the contact time in milliseconds, which makes the force come out a thousand times too small.
- Reading F_avg as the peak force. The real force spikes well above the average during contact and falls back to zero at both ends.
- Forgetting that impulse points the same way as the momentum change, not the same way as the object's motion.
Related formulas
- Work-energy theorem:
W_net = ΔKE = ½mv² - ½mv₀² - Momentum formula:
p = mv - Newton's second law:
ΣF = ma